Concentration analysis of multivariate elliptic diffusion processes

TL;DR

This paper establishes new concentration inequalities and PAC bounds for additive functionals of multivariate diffusion processes, enabling rigorous analysis in high-dimensional and sampling contexts.

Contribution

It introduces a broad approach via the Poisson equation to derive concentration bounds for unbounded functions of nonreversible diffusions, extending previous results.

Findings

Validated restricted eigenvalue condition for Lasso in high dimensions

Provided PAC bounds for Langevin MCMC sampling of heavy-tailed densities

Extended concentration inequalities to a wider class of unbounded functions

Abstract

We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the power of these exponential inequalities by two examples of very different areas. Considering a possibly high-dimensional parametric nonlinear drift model under sparsity constraints, we apply the continuous-time concentration results to validate the restricted eigenvalue condition for Lasso estimation, which is fundamental for the derivation of oracle inequalities. The results for discrete additive functionals are used to investigate the unadjusted Langevin MCMC algorithm for sampling of moderately heavy-tailed densities $\pi$. In particular, we provide PAC bounds for the sample Monte Carlo estimator of integrals $\pi(f)$ for polynomially growing functions $f$ that quantify sufficient sample and step sizes for approximation within a prescribed margin with high probability.